In recent years, with the appearance of “turbo principle”, iterative receivers are becoming more and more popular and promising because of their excellent performances. Different mechanisms have been proposed and studied, for example, iterative detection, iterative multi-input multi-output (MIMO) equalization, etc.
However, these iterative mechanisms are seriously affected by channel estimator. For example, it has been shown that an iterative MIMO equalizer is more sensitive to channel estimation, and the traditional non-iterative channel estimators cannot provide sufficiently accurate channel estimates.
This necessitates more accurate channel estimates in order to improve system performances. Recently, iterative channel estimation is being considered to improve the accuracy of channel estimation, which uses the “soft” information of data to improve channel estimation performance. This type of channel estimation algorithms is particularly helpful for systems which have fewer and/or lower powered pilot symbols. For example, in Long Term Evolution (LTE) systems, at most two (2) OFDM symbols carry pilots in a given resource block and this decreases to one (1) OFDM symbol for MIMO transmission. A resource block (RB) is the minimum allocation unit over seven (7) OFDM symbols and twelve (12) subcarriers.
With this sparse pilot arrangement, the iterative channel estimation can be a good candidate to improve channel estimates. Moreover, for future standards, one of the key features is to build more power efficient transmission systems and, in this manner, decreasing the power of pilots is one of the possible ways to improve the power consumption efficiency. In such systems, the channel estimation algorithms used in current systems will have less accuracy and more robust algorithms will be needed.
Some iterative channel estimators have already been proposed for orthogonal frequency-division multiplexing (OFDM) systems by using the extrinsic information from decoder. Among these iterative algorithms, the expectation maximization (EM) based channel estimation is taking attention because of its attractive performance. The EM algorithm is an iterative method to find the maximum-likelihood (ML) estimates of parameters in the presence of unobserved data. The idea behind the algorithm is to augment the observed data with latent data, which can be either missing data or parameter values, so that the likelihood function conditioned on the data and the latent data has a form that is easy to manipulate.
In others words, instead of computing the maximum-likelihood (ML) channel estimate from the observations only, the EM algorithm makes use of the so-called complete data κ, which are not observed directly but only through incomplete data.
The EM algorithm performs a two-step procedure: the “E-step” and the “M-step”. In the case of an estimation of a channel, these two steps have the following form:
1) E-step: compute the auxiliary function:Q(h|ĥ(i))=Eκ└log p(κ|h)|Y, ĥ(i)┘;  (1)2) M-step: update the parameters:
                                                        h              ^                                      (                              i                +                1                            )                                =                      arg            ⁢                                                  ⁢                                          max                h                            ⁢                              Q                (                                  h                  |                                                            h                      ^                                                              (                      i                      )                                                                      )                                                    ,                            (        2        )            
where h stands for the parameters to be estimated; ĥ(i) represents the estimated parameters in the ith iteration; Y stands for the observed data and κ is the so-called “complete data”, which contains observed data and some missed data. Likelihood increases along EM iterations.
In previous works, the EM channel estimation has been proposed for uncoded and coded OFDM systems with the assumption that pilots exist in every OFDM symbol. However, in practical specifications such as 3GPP LTE, IEEE 802.16m and LTE-Advanced, pilot symbols are present on certain OFDM symbols only.
Furthermore, even though the EM channel estimation method provides good performances and convergence property, it has a non negligible complexity because of a matrix inversion. When it is fully performed in one LTE sub-frame, the complexity of channel estimation and the latency are raised.
In a practical system like LTE, the traditional EM channel estimation method has another problem: it always considers the whole bandwidth, which is not the case in practical systems. Furthermore, in a practical system, “null” (guard) subcarriers are inserted at both sides of the bandwidth.
These “null” subcarriers make the traditional EM algorithm diverge, which is a substantial problem and leads to an inefficiency of the method.
Thus, it is important to propose an EM algorithm for practical systems and to make it converge even in the presence of “null” subcarriers. There is furthermore a need for proposing an EM channel estimation method in which the complexity of the calculation is reduced in order to shorten latency time.